Observation any continuous time signal can be expressed as su perposition of

# Observation any continuous time signal can be

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Observation: any continuous-time signal can be expressed as su- perposition of infinitesimally time–shifted continuous-time unit im- pulses — Sampling or sifting property of unit impulse x ( t ) = integraldisplay -∞ x ( τ ) δ ( t - τ ) Example: Unit step function u ( t ) = t integraldisplay -∞ δ ( τ ) = integraldisplay 0 δ ( t - τ ) Lampe, Schober: Signals and Communications
39 squaresolid LTI system characterization Define δ ( t - τ ) -→ h τ ( t ) Linearity integraldisplay -∞ x ( τ ) δ ( t - τ ) = x ( t ) -→ y ( t ) = integraldisplay -∞ x ( τ ) h τ ( t ) Time invariance δ ( t - τ ) -→ h τ ( t ) = h 0 ( t - τ ) Impulse response h ( t ) h 0 ( t ) Convolution integral y ( t ) = integraldisplay -∞ x ( τ ) h ( t - τ ) squaresolid Symbolically representation of convolution y ( t ) = x ( t ) * h ( t ) Lampe, Schober: Signals and Communications
40 Example: Input: x ( t ) = e - at u ( t ) , Re { a } > 0 Impulse response: h ( t ) = u ( t ) Output: y ( t ) = ? Convolution integral y ( t ) = integraldisplay -∞ x ( τ ) h ( t - τ ) Simplify ( x ( t ) = 0 for t < 0 ) y ( t ) = integraldisplay 0 x ( τ ) h ( t - τ ) Causal system ( h ( t - τ ) = 0 for t - τ < 0 ) y ( t ) = t integraldisplay 0 x ( τ ) h ( t - τ ) t 0 : y ( t ) = t integraldisplay 0 e - = - 1 a e - vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle t 0 = 1 a (1 - e - at ) For all t y ( t ) = 1 a (1 - e - at ) u ( t ) Lampe, Schober: Signals and Communications
41 squaresolid Graphical interpretation Time-reverse h ( τ ) h ( - τ ) Shift h ( - τ ) by t h ( t - τ ) Integrate products of overlapping components Example: * Input: x ( t ) = braceleftbigg 1 , 0 t 2 T 0 , otherwise Impulse response: h ( t ) = 2 , 0 t < T 3 , T t 2 T 0 otherwise * t < 0 t - 2 T 2 T T t h ( t - τ ) x ( τ ) τ No overlap: y ( t ) = 0 * 0 t < T t T 2 T τ y ( t ) = t integraldisplay 0 1 · 2 dt = 2 t Lampe, Schober: Signals and Communications
42 * T t < 2 T t T 2 T τ y ( t ) = t - T integraldisplay 0 3 dt + t integraldisplay t - T 2 dt = 3( t - T ) + 2 T * 2 T t < 3 T t T 2 T τ y ( t ) = 2 T integraldisplay t - T 2 dt + t - T integraldisplay t - 2 T 3 dt = 2(3 T - t ) + 3 T * 3 T t < 4 T t T 2 T τ y ( t ) = 2 T integraldisplay t - 2 T 3 dt = 3(4 T - t ) Lampe, Schober: Signals and Communications
43 * t 4 T t T 2 T τ No overlap: y ( t ) = 0 * In summary y ( t ) = 0 , t < 0 2 t, 0 t < T 2 T + 3( t - T ) , T t < 2 T 2(3 T - t ) + 3 T, 2 T t < 3 T 3(4 T - t ) , 3 T t < 4 T 0 , t 4 T 5 4 3 2 1 y ( t ) /T 3 4 2 1 t/T Lampe, Schober: Signals and Communications
44 2.3 Properties of Linear Time–Invariant Systems squaresolid Linearity and time invariance complete characterization by im- pulse response y [ k ] = summationdisplay k = -∞ x [ k ] h [ n - k ] = x [ k ] * h [ k ] y ( t ) = integraldisplay -∞ x ( τ ) h ( t - τ ) = x ( t ) * h ( t ) squaresolid Further properties based on and in terms of impulse response repre- sentation 2.3.1 Commutative, Distributive, and Associative Property of Convolution — Interconnections of LTI Systems squaresolid Commutative property: order of the signals to be convolved can be changed Continuous–time case x ( t ) * h ( t ) = h ( t ) * x ( t ) = integraldisplay -∞ h ( τ ) x ( t - τ ) Discrete–time case x [ n ] * h [ n ] = h [ n ] * x [ n ] = summationdisplay k = -∞ h [ k ] x [ n - k ]

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